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Description of Individual Course Units

Course Unit Code Course Unit Title Type Of Course D U L ECTS MAT 5023 ELLIPTIC CURVES ELECTIVE 3 0 0 7

Offered By

Graduate School of Natural and Applied Sciences

Level of Course Unit

Second Cycle Programmes (Master's Degree)

Course Coordinator

Offered to

Mathematics Mathematics

Course Objective

The aim of this course is to introduce the fundamental concepts in elliptic curves, and show its usage in the whole body of mathematics.

Learning Outcomes of the Course Unit

1   Will be able to use affine and projective geometry. 2   Will be able to use rational points on curves. 3   Will be able to use elliptic curves in number theory. 4   Will be able to use height functions. 5   Will be able to use Diophantine equations.

Mode of Delivery

Face -to- Face

Prerequisites and Co-requisites

None

Recomended Optional Programme Components

None

Course Contents

Week Subject Description 1 Affine and projective spaces, lines in the projective space 2 Conics and their rational points 3 Geometry of elliptic curves 4 Elliptic curves over various fields 5 The group law and computations 6 Torsion points and Nagell-Lutz theorem 7 Absolute values on number fields 8 Midterm 9 Height functions 10 Mordell-Weil theorem and the rank of elliptic curves 11 Congruent number problem 12 Integer points on elliptic curves 13 L-functions of elliptic curves, connections to modular forms 14 Connections to Fermat last theorem

Recomended or Required Reading

Textbook(s): [1] Silverman, Joseph H., Tate, John T., Rational Points on Elliptic Curves, second edition Springer. [2] Silverman, Joseph H, Arithmetic of elliptic curves, second edition, Springer. References: Materials: Instructor s notes and presentations

Planned Learning Activities and Teaching Methods

Lecture notes Presentation Problem solving

Assessment Methods

SORTING NUMBER SHORT CODE LONG CODE FORMULA 1 MTE MIDTERM EXAM 2 ASG ASSIGNMENT 3 RPT REPORT 4 FIN FINAL EXAM 5 FCG FINAL COURSE GRADE MTE * 0.30 +ASG +RPT/2 * 0.30 +FIN * 0.40 6 RST RESIT 7 FCGR FINAL COURSE GRADE (RESIT) MTE * 0.30 +ASG +RPT/2 * 0.30 +RST * 0.40 *** Resit Exam is Not Administered in Institutions Where Resit is not Applicable.

Further Notes About Assessment Methods

None

Assessment Criteria

To be announced.

Language of Instruction

English

Course Policies and Rules

To be announced.

Contact Details for the Lecturer(s)

haydar.goral@deu.tr

Office Hours

To be announced.

Work Placement(s)

None

Workload Calculation

Activities Number Time (hours) Total Work Load (hours) Lectures 13 3 39 Preparations before/after weekly lectures 13 3 39 Preparation for midterm exam 1 15 15 Preparation for final exam 1 25 25 Preparing assignments 10 5 50 Final 1 3 3 Midterm 1 3 3 TOTAL WORKLOAD (hours) 174

Contribution of Learning Outcomes to Programme Outcomes

PO/LO PO.1 PO.2 PO.3 PO.4 PO.5 PO.6 PO.7 PO.8 PO.9 PO.10 PO.11 LO.1 4 3 3 2 2 2 4 3 3 3 LO.2 4 3 3 2 2 2 4 3 3 3 LO.3 4 3 3 2 2 2 4 3 3 3 LO.4 4 3 3 2 2 2 4 3 3 3 LO.5 4 3 3 2 2 2 4 3 3 3